The Postulates What is a postulate? 1 suggest or assume the - - PowerPoint PPT Presentation

The Postulates

What is a postulate?

1 suggest or assume the existence, fact, or truth of (something) as a

basis for reasoning, discussion, or belief. ”a theory postulated by a respected scientist” synonyms: suggest, advance, posit, hypothesize, propose, assume

2 (in ecclesiastical law) nominate or elect (someone) to an ecclesiastical

• ffice subject to the sanction of a higher authority.

How do you know it’s correct? DATA! See here for an example of impeccable logic.

Jerry Gilfoyle The Rules of the Quantum Game 1 / 20

The Postulates (the Rules of the Game)

1

Each physical, measurable quantity, A, has a corresponding operator, ˆ A , that satisfies the eigenvalue equation ˆ A φ = aφ and measuring that quantity yields the eigenvalues of ˆ A .

2

Measurement of the observable A leaves the system in a state that is an eigenfunction of ˆ A .

3

The state of a system is represented by a wave function Ψ that is continuous, differentiable and contains all possible information about the system. The ‘intensity’ is proportional to |Ψ|2 and is interpreted as a probability. The average value of any observable A is A =

• all space Ψ∗ ˆ

A Ψd r.

4

The time and spatial dependence of Ψ(x, t) is determined by the time dependent Schroedinger equation. i ∂ ∂t Ψ(x, t) = − 2 2µ ∂2 ∂x2 Ψ(x, t) + V (x)Ψ(x, t) µ ≡ reduced mass.

Jerry Gilfoyle The Rules of the Quantum Game 2 / 20

Apply the Rules: A Particle in a Box

Consider the infinite rectangular well potential shown in the figure below. What is the time-independent Schroedinger equation for this potential? What is the general solution to the previous question? What are the boundary conditions the solution must satisfy? What is the particular solution for this potential? What is the energy of the particular solution?

a x V(x)

Jerry Gilfoyle The Rules of the Quantum Game 3 / 20

The Infinite Rectangular Well Potential - Energy Levels

1 2 3 4 5 6 7 8 9 10

Energy Energy Levels n

Jerry Gilfoyle The Rules of the Quantum Game 4 / 20

Some Math

The solutions of the Schroedinger equation form a Hilbert space.

1 They are linear, i.e. superposition/interference is built in.

If a is a constant and φ(x) is an element of the space, then so is aφ(x). If φ1(x) and φ2(x) are elements, then so is φ1(x) + φ2(x).

2 An inner product is defined and all elements have a norm.

φn|ψ = ∞

−∞

φ∗

nψ dx

and φn|φn = ∞

−∞

φ∗

nφndx = 1

3 The solutions are complete.

|ψ =

∞

• n=0

bn|φn

4 The solutions are orthonormal so φn|φn′ = δn,n′.

The operators are Hermitian - their eigenvalues are real.

Jerry Gilfoyle The Rules of the Quantum Game 5 / 20

Apply the Rules More: A Particle in a Box

Consider the infinite rectangular well potential shown in the figure below with an initial wave packet defined in the following way. Ψ(x, 0) =

1 √ d

x0 < x < x1 and d = x1 − x0 =

• therwise

What possible values are

• btained in an energy

measurement? What eigenfunctions contribute to this wave packet and what are their probabilities? What will many measurements

• f the energy give?

a x x

1

x V(x) (x) Ψ

Jerry Gilfoyle The Rules of the Quantum Game 6 / 20

Apply the Rules More: A Particle in a Box

Consider the infinite rectangular well potential shown in the figure below with an initial wave packet defined in the following way. Ψ(x, 0) =

1 √ d

x0 < x < x1 and d = x1 − x0 =

• therwise

What possible values are

• btained in an energy

measurement? What eigenfunctions contribute to this wave packet and what are their probabilities? What will many measurements

• f the energy give?

a x x

1

x V(x) (x) Ψ

1 2 3 4 5 6 7 8 9 10

Energy Energy Levels n Jerry Gilfoyle The Rules of the Quantum Game 6 / 20

L’Hˆ

• pital’s Rule

If lim

x→c f (x) = lim x→c g(x) = 0

• r

lim

x→c f (x) = lim x→c g(x) = ±∞

and lim

x→c

f ′(x) g′(x) exists, then lim

x→c

f (x) g(x) = lim

x→c

f ′(x) g′(x) .

Jerry Gilfoyle The Rules of the Quantum Game 7 / 20

Truncated Fourier Series - 1

1 term 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability Jerry Gilfoyle The Rules of the Quantum Game 8 / 20

Truncated Fourier Series - 1

1 term 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability 5 terms 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability Jerry Gilfoyle The Rules of the Quantum Game 8 / 20

Truncated Fourier Series - 1

1 term 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability 5 terms 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability 25 terms 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability Jerry Gilfoyle The Rules of the Quantum Game 8 / 20

Truncated Fourier Series - 1

1 term 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability 5 terms 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability 25 terms 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability 125 terms 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability Jerry Gilfoyle The Rules of the Quantum Game 8 / 20

Truncated Fourier Series - 2

625 terms 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability Jerry Gilfoyle The Rules of the Quantum Game 9 / 20

Truncated Fourier Series - 2

625 terms 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability 1250 term s 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 Position Probability Jerry Gilfoyle The Rules of the Quantum Game 9 / 20

Probabilities of Different Final States

a = 1.0 Å x0 = 0.3 Å x1 = 0.5 Å 500 1000 1500 2000 2500 3000 0.0 0.1 0.2 0.3 0.4 Energy (eV) Probability Rectangular Wave in a Square Well

Jerry Gilfoyle The Rules of the Quantum Game 10 / 20

Probabilities of Different Final States - 2

a = 1.0 Å x0 = 0.3 Å x1 = 0.5 Å 2000 4000 6000 8000 10000 12000 14000 10-4 0.001 0.010 0.100 1 Energy (eV) log(Probability) Rectangular Wave in a Square Well Jerry Gilfoyle The Rules of the Quantum Game 11 / 20

Probabilities of Different Final States - 3

20 40 60 80 100 10-5 10-4 0.001 0.010 0.100 1 Energy Level log(Probability) Rectangular Wave in a Square Well

a = 1.0 ˚ A x0 = 0.3 ˚ A x1 = 0.5 ˚ A

Jerry Gilfoyle The Rules of the Quantum Game 12 / 20

Probabilities of Different Final States - 4

20 40 60 80 100 10-5 10-4 0.001 0.010 0.100 1 Energy Level log(Probability) Rectangular Wave in a Square Well

Jerry Gilfoyle The Rules of the Quantum Game 13 / 20

Why a page limit?

Nature 129, 312 (27 February 1932) Jerry Gilfoyle The Rules of the Quantum Game 14 / 20

Width of a distribution?

50 100 150 200 0.00 0.05 0.10 0.15 0.20 kn (a.u.) |bn

2 Jerry Gilfoyle The Rules of the Quantum Game 15 / 20

The Quantum Program in One Dimension - So Far

1

Solve the Schroedinger equation to get eigenfunctions and eigenvalues. − 2 2m ∂2φ(x) ∂x2 + V φ(x) = Enφ(x)

2

For an initial wave packet ψ(x) use the completeness of the eigenfunctions. |ψ(x) =

∞

• n=1

bn|φ(x)

3

Apply the orthonormality φm|φn = δm,n. φm|ψ = φm| ∞

• n=1

bn|φ

• = bm =

∞

−∞

φ∗

m

∞

• n=1

bn|φ

• dx

4

Get the probability Pn for measuring En from |ψ. Pn = |bn|2

5

Do the free particle solution.

6

Put in the time evolution.

Jerry Gilfoyle The Rules of the Quantum Game 16 / 20

Nuclear Fusion!

Consider a case of one dimensional nuclear ‘fusion’. A neutron is in the potential well of a nucleus that we will approximate with an infinite square well with walls at x = 0 and x = a. The eigenfunctions and eigenvalues are En = n22π2 2ma2 φn =

• 2

a sin nπx a

• 0 ≤ x ≤ a

= x < 0 and x > a . The neutron is in the n = 4 state when it fuses with another nucleus that is the same size, instantly putting the neutron in a new infinite square well with walls at x = 0 and x = 2a.

1 What are the new eigenfunctions and eigenvalues of the fused system? 2 What is the spectral distribution? 3 What is the average energy? Use the bn’s. Jerry Gilfoyle The Rules of the Quantum Game 17 / 20

Spectral Distribution for One-Dimensional Nuclear Fusion

50 100 150 200 0.0 0.1 0.2 0.3 0.4 0.5 0.6 En (units of E1) |bn

2

Nuclear Fusion

Jerry Gilfoyle The Rules of the Quantum Game 18 / 20

Spectral Distribution for One-Dimensional Nuclear Fusion

200 400 600 800 1000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 En (units of E1) |bn

2

Nuclear Fusion

Jerry Gilfoyle The Rules of the Quantum Game 19 / 20

Spectral Distribution for One-Dimensional Nuclear Fusion

Only non-zero values bn=0 for n even, except n=8 200 400 600 800 1000 10-5 10-4 0.001 0.010 0.100 1 En (units of E1) |bn

2

Nuclear Fusion

Jerry Gilfoyle The Rules of the Quantum Game 20 / 20